If you can build your own computer (It's not hard, and there are like 1000 FAQS online for doing it that are pretty straightforward), then I suggest you do that.
I put together a decent machine for around 500 about 6 months ago. Aside from the explosion that was my power supply that one day, everything else about the system was great, and my replacement power supply was even better.
I would suggest a mid-grade motherboard, nothing too flashy but nothing cheap, and also a graphics card that basically was the shit like 1 to 2 years ago. Which means now it will be able to moderately run the games of today. However you could save up and upgrade that later.
Also you may not want a 1 TB HD, but they're pretty cheap now, and think of all the piracy you can do
As far as dual core vs. quad core, there's a lot of debate in that area, but it just depends on the situation really. Check the reviews of the various processors, see what people are saying about them. I mean I doubt you're going to get into anything too rough like over-clocking etc. anyway.
Also I would highly suggest at least 4 gb of ram if you plan on running Windows 7. You can do it with 2, but it's so much smoother with 4.
Re: New Computer advice.
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